A Sobolev-type upper bound for rates of approximation by linear combinations of Heaviside plane waves

Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f ∈ Cd(R), with lower-order partials vanishing at infinity and dth-order partials vanishing as ‖x‖−(d+1+ε), ε > 0, on any domain ⊂ R with unit Lebesgue measure, the L2( )-error in approximating f by a linear combination of n Heaviside plane waves is bounded above by kd‖f ‖d,1,∞n−1/2, where kd ∼ ( d)1/2(e/2 )d/2 and ‖f ‖d,1,∞ is the Sobolev seminorm determined by the largest of the L1-norms of the dth-order partials of f onR . In particular, for d odd and f (x)=exp(−‖x‖2), the L2( )-approximation error is at most (2 d)3/4n−1/2 and the sup-norm approximation error onR is at most 68 √ 2(n−1)−1/2(2 d)3/4 √ d + 1, n 2. © 2007 Published by Elsevier Inc.